2021/12/01 by Juntao Duan, Duan, Juntao, Popescu, Ionel +1
Mathematics · #Advanced Algebra and Geometry #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Mathematical Analysis and Transform Methods #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2112.00300
openalex publication_date 2021/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when project high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random matrix affect norms of random vectors. In particular we prove the distribution of the norm of random vector X ∈ ℝn, whose entries are i.i.d. random variables, is preserved by random projection S:ℝn → ℝm. More precisely, (XTSTSX - mn)/(√(σ2 m2n+2mn2)) \xrightarrow[ m/n→ 0 ] m,n→ ∞ N(0,1) We also prove a concentration of the random norm transformed by either random projection or random embedding. Overall, our results showed random matrix has low distortion for the norm of random vectors with i.i.d. entries.